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In the plane, the locus of the points having the ratio of their distances from a certain point (the focus) and from a certain line (the directrix) equal to a given constant εε\varepsilon, is a conic section, which is an ellipse, a parabola or a hyperbola depending on whether εε\varepsilon is less than, equal to or greater than 1.

For showing this, we choose the yyy-axis as the directrix and the point (q, 0)q 0(q,\,0) as the focus. The locus condition reads then

Place the origin into a focus of a conic section (and in the cases of ellipse and hyperbola, the abscissa axis through the other focus). As before, let qqq be the distance of the focus from the corresponding directrix. Let rrr and φφ\varphi be the polar coordinates of an arbitrary point of the conic. Then the locus condition may be expressed as